There have widely been used multitube heat exchangers having a tubular housing including heads defined by partitions in the opposite sides of the tubular housing. One of the heads has a fluid inlet port and the other head has a fluid outlet port. The housing accommodates a number of pipes juxtaposed between the heads for passing a fluid therethrough. The housing has fluid inlet and outlet holes for passing another fluid around the tubes, so that heat can be exchanged between the fluids through the walls of the tubes. If a multitube heat exchanger of this type is reduced in size for use as a heat exchanger for an artificial lung or the like, for example, the volumes of the heads are relatively large as compared with the effective heat-exchanging area of the heat exchanger. As a result, the speed of flow of the fluid in the pipes is greatly affected by the position of the pipes because of the directivity of the fluid flowing into one of the heads. Particularly, if the inlet port of the head is oriented tangentially to the tubular housing, the fluid produces a swirling motion in the head due to its directivity, thus developing a pressure difference between the central and outer circumferential regions in the head. Therefore, the speed of flow of the fluid flowing in those pipes which are located in the central region largely differs from the speed of flow of the fluid flowing in those pipes which are located in the outer circumferential region, with the result that the temperature in the heat exchanger becomes ununiform. The fluid in the central pipes does not flow, and hence a heat exchange between the fluid in the central pipes and the other fluid flowing around the central pipes is ineffective, and hence the efficiency of the entire heat exchanger is low. It is thus difficult to control the temperature of the other fluid in its entirety. Air bubbles introduced into the inlet port of the head tend to be attracted to the central lowerpressure region in the head. Consequently, these air bubbles are liable not to be discharged out through the pipes, making the temperature in the heat exchanger more uneven.